Page:Aether and Matter, 1900.djvu/211

and also

${\frac {df_{1}}{dt_{1}}}={\frac {df}{dt}};$

hence ${\frac {df}{dx'}}+{\frac {dg}{dy'}}+{\frac {dh}{dz'}}-{\frac {v}{c^{2}}}\left({\frac {df}{dt'}}-v{\frac {df}{dx'}}\right)$ is equal to

$\epsilon {\frac {df_{1}}{dx_{1}}}+{\frac {dg_{1}}{dy_{1}}}+{\frac {dh_{1}}{dz_{1}}}-{\frac {v}{c^{2}}}\epsilon {\frac {df}{dt}},$

so that, up to the order of $(v/c)^{2}$ inclusive,

${\frac {df}{dx'}}+{\frac {dg}{dy'}}+{\frac {dh}{dz'}}={\frac {df_{1}}{dx_{1}}}+{\frac {dg_{1}}{dy_{1}}}+{\frac {dh_{1}}{dz_{1}}}.$

Thus the conclusions as to the corresponding positions of the electrons of the two systems, which had been previously established up to the first order of v/c, are true up to the second order when the dimensions of the moving system are contracted in comparison with the fixed system in the ratio $\epsilon ^{-{\frac {1}{2}}}$ or $1-{\frac {1}{2}}v^{2}/c^{2}$, along the direction of its motion.

111. The ratio of the strengths of corresponding electrons in the two systems may now be deduced just as it was previously when the discussion was confined to the first order of v/c. For the case of a single electron in uniform motion the comparison is with a single electron at rest, near which $(a_{1},\ b_{1},\ c_{1})$ vanishes so far as it depends on that electron: now we have in the general correlation

$g=g_{1}+{\frac {v}{4\pi c^{2}}}(c_{1}+4\pi vg),$

hence in this particular case

$(g,\ h)=\epsilon (g_{1},\ h_{1}),$ while $f=\epsilon ^{\frac {1}{2}}f_{1}.$

But the strength of the electron in the moving system is the value of the integral $\int \int (f\ dy'\ dz'+g\ dz'\ dx'+h\ dx'\ dy')$ extended over any surface closely surrounding its nucleus; that is here $\epsilon ^{\frac {1}{2}}\int \int \left(f_{1}dy_{1}dz_{1}+g_{1}dz_{1}dx_{1}+h_{1}dx_{1}dy_{1}\right)$, so that the strength of each moving electron is $\epsilon ^{\frac {1}{2}}$ times that of the correlative fixed electron. As before, no matter what other electrons are present, this